{
  "id": "1711.03868",
  "version": "v1",
  "published": "2017-11-09T14:05:01.000Z",
  "updated": "2017-11-09T14:05:01.000Z",
  "title": "On the $A_α$-characteristic polynomial of a graph",
  "authors": [
    "Xiaogang Liu",
    "Shunyi Liu"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1709.00792",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G) $$ for any real $\\alpha\\in [0,1]$. The $A_{\\alpha}$-characteristic polynomial of $G$ is defined to be $$ \\det(xI_n-A_{\\alpha}(G))=\\sum_jc_{\\alpha j}(G)x^{n-j}, $$ where $\\det(*)$ denotes the determinant of $*$, and $I_n$ is the identity matrix of size $n$. The $A_{\\alpha}$-spectrum of $G$ consists of all roots of the $A_{\\alpha}$-characteristic polynomial of $G$. A graph $G$ is said to be determined by its $A_{\\alpha}$-spectrum if all graphs having the same $A_{\\alpha}$-spectrum as $G$ are isomorphic to $G$. In this paper, we first formulate the first four coefficients $c_{\\alpha 0}(G)$, $c_{\\alpha 1}(G)$, $c_{\\alpha 2}(G)$ and $c_{\\alpha 3}(G)$ of the $A_{\\alpha}$-characteristic polynomial of $G$. And then, we observe that $A_{\\alpha}$-spectra are much efficient for us to distinguish graphs, by enumerating the $A_{\\alpha}$-characteristic polynomials for all graphs on at most 10 vertices. To verify this observation, we characterize some graphs determined by their $A_{\\alpha}$-spectra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-09T14:05:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "characteristic polynomial",
      "adjacency matrix",
      "degree matrix",
      "identity matrix",
      "first formulate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}